Finite groups with only small automorphism orbits

Abstract

<jats:title>Abstract</jats:title><jats:p>We study finite groups <jats:italic>G</jats:italic> such that the maximum length of an orbit of the natural action of the automorphism group <jats:inline-formula id="j_jgth-2019-0152_ineq_9999_w2aab3b7e2701b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0152_eq_0461.png" /><jats:tex-math>{\mathrm{Aut}(G)}</jats:tex-math></jats:alternatives></jats:inline-formula> on <jats:italic>G</jats:italic> is bounded from above by a constant. Our main results are the following: Firstly, a finite group <jats:italic>G</jats:italic> only admits <jats:inline-formula id="j_jgth-2019-0152_ineq_9998_w2aab3b7e2701b1b6b1aab1c15b1b9Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0152_eq_0461.png" /><jats:tex-math>{\mathrm{Aut}(G)}</jats:tex-math></jats:alternatives></jats:inline-formula>-orbits of length at most 3 if and only if <jats:italic>G</jats:italic> is cyclic of one of the orders 1, 2, 3, 4 or 6, or <jats:italic>G</jats:italic> is the Klein four group or the symmetric group of degree 3. Secondly, there are infinitely many finite (2-)groups <jats:italic>G</jats:italic> such that the maximum length of an <jats:inline-formula id="j_jgth-2019-0152_ineq_9997_w2aab3b7e2701b1b6b1aab1c15b1c17Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0152_eq_0461.png" /><jats:tex-math>{\mathrm{Aut}(G)}</jats:tex-math></jats:alternatives></jats:inline-formula>-orbit on <jats:italic>G</jats:italic> is 8. Thirdly, the order of a <jats:italic>d</jats:italic>-generated finite group <jats:italic>G</jats:italic> such that <jats:italic>G</jats:italic> only admits <jats:inline-formula id="j_jgth-2019-0152_ineq_9996_w2aab3b7e2701b1b6b1aab1c15b1c27Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0152_eq_0461.png" /><jats:tex-math>{\mathrm{Aut}(G)}</jats:tex-math></jats:alternatives></jats:inline-formula>-orbits of length at most <jats:italic>c</jats:italic> is explicitly bounded from above in terms of <jats:italic>c</jats:italic> and <jats:italic>d</jats:italic>. Fourthly, a finite group <jats:italic>G</jats:italic> such that all <jats:inline-formula id="j_jgth-2019-0152_ineq_9995_w2aab3b7e2701b1b6b1aab1c15b1c37Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2019-0152_eq_0461.png" /><jats:tex-math>{\mathrm{Aut}(G)}</jats:tex-math></jats:alternatives></jats:inline-formula>-orbits on <jats:italic>G</jats:italic> are of length at most 23 is solvable.</jats:p>

Finite transitive permutation groups with only small normaliser orbits

“…The goal of this paper is to extend the main results of [5] to Aut perm (G). By a classical idea, dating back to Cayley, abstract groups are in a bijective correspondence with regular permutation groups via their (right) multiplication actions on themselves.…”

“…In his recent paper [5], the first author studied finite groups that are at the opposite end of the spectrum, i.e., they are "highly unsymmetrical". Formally, the paper [5] is concerned with finite groups G such that all orbits of Aut(G) on G have size bounded by a constant.…”

“…In his recent paper [5], the first author studied finite groups that are at the opposite end of the spectrum, i.e., they are "highly unsymmetrical". Formally, the paper [5] is concerned with finite groups G such that all orbits of Aut(G) on G have size bounded by a constant. It should be noted that Robinson and Wiegold already studied such conditions for general groups in their 1984 paper [25], obtaining inter alia a nice general structural characterisation [25,Theorem 1] in the spirit of Neumann's celebrated characterisation of BFC-groups [24,Theorem 3.1].…”

“…It should be noted that Robinson and Wiegold already studied such conditions for general groups in their 1984 paper [25], obtaining inter alia a nice general structural characterisation [25,Theorem 1] in the spirit of Neumann's celebrated characterisation of BFC-groups [24,Theorem 3.1]. However, the methods and results from [5] are tailored to the finite case, where more specific statements can be made.…”

“…Moreover, if G is a regular permutation group, then Aut perm (G) = Aut(G) (see Lemma 2.2.3 below), and so the statements of [5, Theorem 1.1] may equivalently be viewed as results on finite regular permutation groups G such that all Aut perm (G)-orbits on G have size bounded by a constant. In this paper, we will extend [5,Theorem 1.1] from finite regular to finite transitive permutation groups. Our main results are as follows:…”